Roger Penrose

Born: 8 August 1931 in Colchester, Essex, England

Roger Penrose's parents, Lionel Sharples Penrose and Margaret Leathes, were both medically trained. Margaret was a doctor while Lionel was a medical geneticist who was elected a Fellow of the Royal Society. He was involved with a project called the Colchester survey which aimed to discover whether inherited factors or environmental factors were the most significant in determining if someone would be likely to suffer from mental heath problems. He was in Colchester carrying out this work at the time Roger was born. Roger's brother, Oliver Penrose, had been born two years earlier. Oliver went on to become professor of mathematics first at the Open University, then at Heriot-Watt University in Edinburgh, Scotland. Roger also had a younger brother Jonathan who went on to become a lecturer in psychology. Jonathan was British Chess Champion ten times between 1958 and 1969 and, many argue, was the most naturally talented British chess player of all time.

In 1939 Roger's father went to the United States with his family but as all the indications pointed towards the outbreak of war, he decided not to return to England with his family but accepted an appointment in a hospital in London, Ontario, Canada. Roger attended school in London, Ontario but although it was during this period that he first became interested in mathematics it was not his schooling which stimulated this interest, rather it was his family. He writes ([2] or [3]):-

I remember making various polyhedra when I was about ten ...

Roger's father became Director of Psychiatric Research at the Ontario Hospital in London Ontario, but he was very interested in mathematics, particularly geometry, while Roger's mother was also interested in geometry. Roger's brother Oliver ([2] or [3]):-

... was two years older than I was, but four years ahead in school. He knew a lot about mathematics at a young age and took a great interest in both mathematics and physics.

In 1945, after the World War II ended, the Penrose family returned to England. Roger's father was appointed as Professor of Human Genetics at University College London and Roger attended University College School in London. Then his interest in mathematics began to increase but his family saw him following in his father's footsteps and taking up a medical career. However, as was typical in schools at this time, biology and mathematics were alternatives at the University College School with pupils having to choose one or the other ([2] or [3]):-

... I remember an occasion when we had to decide which subjects to do in the final two years. Each of us would go up to see the headmaster, one after the other, and he said "Well, what subjects do you want to do when you specialise next year". I said "I'd like to do biology, chemistry and mathematics" and he said "No, that's impossible - you can't do biology and mathematics at the same time, we just don't have that option". Since I had no desire to lose my mathematics I said "Mathematics, physics and chemistry". My parents were rather annoyed when I got home; my medical career had disappeared in one stroke.

Penrose entered University College London which he was entitled to do without paying fees since his father was professor there. He was awarded a B.Sc. degree with First Class Honours in Mathematics and then decided to go to Cambridge to undertake research in pure mathematics. He was following in the footsteps of his older brother Oliver who had also taken his undergraduate degree at University College London and had gone to Cambridge to undertake research but Oliver had chosen physics. Roger, however, was set on research in mathematics and on entering St John's College he began research in algebraic geometry supervised by Hodge. However, after one year of study at Cambridge, finding that his interests were not particularly central to those of Hodge, he changed his supervisor to John Todd. Penrose was awarded his Ph.D. for his work in algebra and geometry from the University of Cambridge in 1957 but by this time he had already become interested in physics. He described how three courses which he attended during his first year at Cambridge influenced him ([2] or [3]):-

I remember going to three courses, none of which had anything to do with the research I was supposed to be doing. One was a course by Hermann Bondi on general relativity which was fascinating ... Another was a course by Paul Dirac on quantum mechanics which was beautiful in a completely different way ... And the third course ... was a course on mathematical logic by Steen. I learnt about Turing machines and Gödel's theorem ...

The first major influence prompting his interest in physics had been Dennis Sciama, a physicist friend of his brother. Penrose said ([2] or [3]):-

[Sciama] was very influential on me. He taught me a great deal of physics, and the excitement of doing physics came through; he was that kind of person, who conveyed the excitement of what was currently going on in physics ...

While at Cambridge working towards his doctorate he began to publish articles on semigroups, and on rings of matrices. In 1955 he published A generalized inverse for matrices in the Proceedings of the Cambridge Philosophical Society. In this paper Penrose defined a generalized inverse X of a complex rectangular (or possibly square and singular) matrix A to be the unique solution to the equations AXA = A, XAX = X, (AX)T = AX, (XA)T = XA. He used this generalized inverse for problems such as solving systems of matrix equations, and finding a new type of spectral decomposition. His second publication of 1955 was A note on inverse semigroups published in the same journal and co-authored with Douglas Munn. An inverse semigroup is a generalisation of a group and continues to be the subject of many research papers. This early paper gave several alternative definitions. In the following year Penrose published On best approximation solutions of linear matrix equations which used the generalized inverse of a matrix to find the best approximate solution X to AX = B where A is rectangular and non-square or square and singular.

Penrose spent the academic year 1956-57 as an Assistant Lecturer in Pure Mathematics at Bedford College, London and was then appointed as a Research Fellow at St John's College, Cambridge. This was a three year post and during its tenure he married Joan Isabel Wedge in 1959. Before the fellowship ended Penrose had been awarded a NATO Research Fellowship which enabled him to spend the years 1959-61 in the United States, first at Princeton and then at Syracuse University. Back in England, Penrose spent the following two years 1961-63 as a Research associate at King's College, London before returning to the United States to spend the year 1963-64 as a Visiting Associate Professor at the University of Texas at Austin.

In 1964 Penrose was appointed as a Reader at Birkbeck College, London and two years later he was promoted to Professor of Applied Mathematics there. In 1973 he was appointed Rouse Ball Professor of Mathematics at the University of Oxford and he continued to hold this until he became Emeritus Rouse Ball Professor of Mathematics in 1998. In that year he was appointed Gresham Professor of Geometry at Gresham College, London.

Beginning in 1959, Penrose published a series of important papers on cosmology. The first was The apparent shape of a relativistically moving sphere while in 1960 he published A spinor approach to general relativity. This latter paper was described as follows:-

An elegant and detailed exposition ... of the mathematical apparatus of gravitation theory, with emphasis on the geometrical theory of the Riemann tensor.

As well as important papers on cosmology, Penrose continues to publish papers on pure mathematics. Together with Henry Whitehead and Christopher Zeeman he published Imbedding of manifolds in euclidean space in 1961. Among other results, the authors prove in this paper that, if 0 < 2m ≤ n, then every closed (m-1)-connected n-manifold can be imbedded in R2n-m+1. This time with Ezra Newman, Penrose published An approach to gravitational radiation by a method of spin coefficients in the following year in which they show that:-

... the two-component spinor formalism leads to the consideration of a tetrad in space-time consisting of two real null-vectors and two complex conjugate ones.

In 1965, using topological methods, Penrose proved an important theorem which, under conditions which he called the existence of a trapped surface, proved that a singularity must occur in a gravitational collapse. Basically under these conditions space-time cannot be continued and classical general relativity breaks down. Penrose looked for a unified theory combining relativity and quantum theory since quantum effects become dominant at the singularity.

One of Penrose's major breakthroughs was his introduction of twistor theory in an attempt to unite relativity and quantum theory. This is a remarkable mathematical theory combining powerful algebraic and geometric methods. Together with Wolfgang Rindler, Penrose published this first volume of Spinors and space-time in 1984. This volume covered two-spinor calculus and relativistic fields while the second volume covering spinor and twistor methods in space-time geometry appeared two years later.

It is for a number of outstanding popular books that Penrose is perhaps best known. He published The Emperor's New Mind : Concerning computers, minds, and the laws of physics in 1989. In the following year the book was awarded the Rhone-Poulenc Science Book Prize. Sklar, reviewing the book, writes that its aim is:-

... to expound and critically attack one recent view of the nature of mind ... taken as reducing mental activity to the carrying out of an algorithmic process, and to propose that a more adequate theory of mind will have to be founded on an as yet not existing physical theory adequate to the known nature of the material world. In the process of the argument elegant expositions, at a level suitable for the unlearned but reasonably sophisticated reader, are given of a wide variety of topics ranging from the nature of algorithms and abstract computability, through results on undecidability and incompleteness, the basic structures of classical physics, the basic structures and philosophical puzzles in quantum mechanics, the basic features of entropic asymmetry and its relation to cosmological structure, the search for an adequate quantum theory of gravity, to some of the results of neuro-anatomy and research into the functioning of the brain.

In 1994 Penrose published Shadows of the mind : A search for the missing science of consciousness which continues to develop the topic of The emperor's new mind. In 1996 Penrose and Hawking published The nature of space and time. This book is a record of a debate between the two at the Isaac Newton Institute of Mathematical Sciences at the University of Cambridge in 1994. Each of the two gave three lectures given alternately so that each could respond to the other's arguments, and then, in a final session, there is a debate between the two. We quote from Penrose's contribution since he states clearly his own position, and that of Hawking:-

At the beginning of this debate Stephen said that he thinks that he is a positivist, whereas I am a Platonist. I am happy with him being a positivist, but I think that the crucial point here is, rather, that I am a realist. Also, if one compares this debate with the famous debate of Bohr and Einstein, some seventy years ago, I should think that Stephen plays the role of Bohr, whereas I play Einstein's role! For Einstein argued that there should exist something like a real world, not necessarily represented by a wave function, whereas Bohr stressed that the wave function doesn't describe a "real" microworld but only "knowledge" that is useful for making predictions.

There is one further aspect of Penrose's work which we must mention. This is his work on non-periodic tilings, an interest which he took up while a graduate student at Cambridge. His first attempts led to success but with a large number of tiles. Further work over many years led to Penrose discovering that he could find non-periodic tilings with only six tiles, then finally he achieved the seemingly impossible with finding non-periodic tilings with only two tiles. By non-periodic we mean that the tilings are not invariant under any translation. Here are some properties of the tiling: in any finite tiled region, only one tiling is possible; in an infinite tiling of the plane, any tiling of a region that occurs is repeated infinitely often elsewhere in the plane and must reoccur within twice the diameter of the region from where you first found it. In fact the tiling of any finite region will eventually appear in every Penrose tiling.

In addition to Penrose's main appointments which we have mentioned above, he also held a number of visiting and part-time posts. He held visiting positions at Yeshiva, Princeton and Cornell during 1966-67 and 1969. From 1983 until 1987 he was Lovett Professor at Rice University in Houston. He then became Distinguished Professor of Physics and Mathematics at Syracuse University in New York until 1993 when he became Francis and Helen Pentz Distinguished Professor of Physics and Mathematics at Pennsylvania State University.

Penrose has received many honours for his contributions. He was elected a Fellow of the Royal Society of London (1972) and a Foreign Associate of the United States National Academy of Sciences (1998). We mentioned the Science Book Prize (1990) which he received for The Emperor's New Mind but this is only one of many prizes. Others include the Adams Prize from Cambridge University; the Wolf Foundation Prize for Physics (jointly with Stephen Hawking for their understanding of the universe): the Dannie Heinemann Prize from the American Physical Society and the American Institute of Physics; the Royal Society Royal Medal; the Dirac Medal and Medal of the British Institute of Physics; the Eddington Medal of the Royal Astronomical Society; the Naylor Prize of the London Mathematical Society; and the Albert Einstein Prize and Medal of the Albert Einstein Society. In 1994 he was knighted for services to science.

In 2000 he received the Order of Merit. He was awarded the De Morgan Medal by the London Mathematical Society in 2004. Part of the citation reads:-

His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.

The Royal Society awarded Penrose their Copley Medal in 2005. The announcement reads:-

Sir Roger Penrose, OM, FRS has been awarded the Royal Society's Copley medal the world's oldest prize for scientific achievement for his exceptional contributions to geometry and mathematical physics. Sir Roger, Emeritus Rouse Ball Professor of Mathematics at the University of Oxford, has made outstanding contributions to general relativity theory and cosmology, most notably for his work on black holes and the Big Bang.

Martin Rees, President of the Royal Society, explained Penrose's exceptional contributions which led to the award:-

Roger has been producing original and important scientific ideas for half a century. His work is characterised by exceptional geometrical and physical insight. He applied new mathematical techniques to Einstein's theory, and led the renaissance in gravitation theory in the 1960s. His novel ideas on space and time and his concept of 'twistors' are increasingly influential. Even his recreations have had intellectual impact: for instance the 'impossible figures' popularised in Escher's artwork, and the never-repeating patterns of 'Penrose tiling'. He has influenced and stimulated a wide public through his lectures, and his best-selling and wide-ranging books.

On receiving the award, Penrose said:-

The award of the Royal Society's Copley Medal came as a complete surprise to me. It is an extraordinary honour, this being the Royal Society's oldest and most distinguished award, first given just 200 years before I was born. I feel most humbled for my name to be added to that enormously distinguished list of previous recipients.

Several universities have awarded Penrose an honorary degree including New Brunswick University (1992), the University of Surrey (1993), the University of Bath (1994), the University of London (1995), the University of Glasgow (1996), Essex University (1996), the University of St Andrews (1997), Santiniketon University (1998), Warsaw University (2005), Katholieke Universiteit Leuven (2005) and the University of York (2006).

Article by:J J O'Connor and E F Robertson

September 2009

MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Penrose.html]